Meticulously studying the interaction between laser fields and atoms has become the core of developing ultra-higher-precision optical clocks. The interaction between laser fields and atoms can cause ac Stark shift, which would affect the accuracy of the measurement of relevant atomic parameters. To reduce the impact of the ac Stark shift, magic-wavelength trapping was introduced in Refs. [1]}, [2]}, [3]}, [4]}. The dynamic electric dipole (\(E1\) ) polarizabilities of a given pair of energy levels are the same as each other in the magic-wavelength trapping, and the second-order Stark shift is eliminated. However, when the accuracy of the optical lattice clock is at the level of \(10^{-18}\) , the multipolar (electric quadrupole \(E2\)  and magnetic dipole \(M1\) ) and nonlinear Stark shifts are non-negligible [5]}, [6]}, [7]}, [8]}, [9]}. These Stark shifts are related to the \(E2\)  and \(M1\)  polarizabilities as well as hyperpolarizability [5]}, [11]}, [7]}, [6]}, [8]}, [15]}, [16]}, [17]}. For pursuing much higher precision, the effects on the systematic uncertainty of optical clocks from the multipolar and nonlinear Stark shifts need to be evaluated [5]}, [6]}, [7]}, [8]}, [15]}, [16]}.
